事件空间中非完整有势系统的参数方程

本文首先在事件空间中建立非完整有势系统的广义方程,然后给出此空间中表示Nielsen算子和Euler算子之间关系的两个定理,最后由这些定理得到广义Nielsen方程。     

变加速动力学系统的广义高斯最小拘束原理

变加速运动在日常生活和工程问题中普遍存在.变加速动力学又称牛顿猝变动力学,因其在混沌理论和非线性动力学中的应用而获得广泛关注.高斯原理是一个具有极值性质的微分变分原理.因此,研究变加速动力学系统的广义高斯原理在理论和应用两方面都有重要意义.文章提出并研究变加速动力学系统的广义高斯原理.首先,引入急动度空间的广义高斯变分概念,将质点的达朗贝尔原理对时间求导数后与广义高斯变分点乘,并利用高斯意义下的理想约束条件,建立了变加速动力学系统的广义高斯原理.在此基础上,通过构造广义拘束函数建立并证明变加速动力学系统的广义高斯最小拘束原理,并给出原理的阿佩尔形式、拉格朗日形式和尼尔森形式.其次,研究原理对变质量力学的推广.从密歇尔斯基方程出发,将它对时间求导并与广义高斯变分点乘,建立了具有理想约束的变质量变加速动力学系统的广义高斯原理.通过构造变质量系统的广义拘束函数,建立并证明变质量力学系统变加速运动的广义高斯最小拘束原理.文中以开普勒-牛顿空间问题为例,利用所得的广义高斯最小拘束原理方法进行计算,验证了方法的有效性.     

Extension of Whittaker equations to non-holonomic mechanical systems

In 1904, using the energy integral Whittaker studied the reduction of a dynamical problem to a problem with fewer degrees of freedom for the holonomic conservative systems and obtained the Whittaker equation^[1].In this article, Whittaker equations are extended to non-holonomic systems and the generalized Whittaker equations are obtained. And then these equations are transformed into Kiel-sen’s form.Finally an example is given.     

A tensor method for the derivation of the equations of rigid body dynamics

A tensor method for the derivation of the equations of rigid body dynamics,based onthe concepts of continuum mechanics,is presented.The formula of time derivative of theinertia tensor with zero corotational rate is used to prove the equivalences of five methods,namely,Lagrange’s equations,Nielsen’s equations,Gibbs-Appell’s equations,Kane’sequations and the generalized momentum type of Kane’s equations.Some differentialidentities on angular velocity and angular acceleration are given.     

Differential equations of motion for a variable-mass holonomic system with a uniformly rotating constraint

In this paper, the differential equations of motion will be established for variable-mass holonomic mechanical systems constrained to the uniform rotation, including the equations in Lagrange form. Nielsen form and Appell form. The application of these new equations is illustrated with an example.     

Nielsen's and Euler's operators of higher order in analytical mechanics

In this paper, the definitions of Nielsen’s and Euler’s operators of higher order are presented. These operators are concerned in analysis for systems with holonomic constraints and non-holonomic constraints of higher order. Some theorems that indicating relation between the two operators are established. Moreover, using the theorems, the new equations of mechanical systems with constraints of higher order are derived. Finally, an example is given.     

APPELL方程和TZNOFF方程在一般非完整系统上的推广

本文给出一个适用于任何阶非完整系统的变分原理。Gauss原理可作为推论由之导出。由此原理出发,将Appell方程和Tzènoff方程推广到一般非完整系统,以一个二阶非完整系统为例,说明所得方程的应用。     

GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Gibbs-Appell’s equations of motion are extended to the most generalvariable mass nonholonomic mechanical systems.Then the Gibbs-Appell’s equations ofmotion in terms of generalized coordinates or quasi-coordinates and an integral variationalprinciple of variable mass nonlinear nonholonomic mechanical systems are obtained.Finally,an example is given.     

On the equations of motion for accelerated frames

In this communication we present the equations of Euler generalized for the motion of a body in an accelerated reference frame using the generalized work-energy principle. The equivalence among the generalized Euler equation, the generalized Lagrange equation, and the generalized Kane equation are shown when applied to the motion of a body of a holonomic system that depend onn generalized coordinates. Therefore when the generalized coordinates can be reduced to two sets of independent coordinates, the generalized Euler equation can be split into two uncoupled equations that are not independent of each other.Universidade da Beira Interior, Covilhã, Portugal. Published in Prikladnaya Mekhanika, Vol. 31, No. 9, pp. 79–89, September, 1995.     

Numerical simulation of nonholonomic rigid-body systems

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D’Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 106–115, September 2006.     

GAUSS PRINCIPLE OF LEAST CONSTRAINT IN GENERALIZED COORDINATES AND ITS GENERALIZATION

当采用广义坐标描述系统的运动时,相比质点形式的高斯最小拘束原理,通过广义坐标形式的高斯最小拘束原理来建立动力学优化模型,计算效率更高. 从高斯原理的变分形式出发推导了广义坐标形式的高斯最小拘束原理,并研究了非理想约束、单边约束及刚体碰撞情形下的高斯最小拘束原理的形式. 研究认为:对刚体碰撞情形下,高斯最小拘束原理不能取代碰撞恢复定律,碰撞恢复定律以碰撞后广义速度的约束方程形式起作用.     

A function theory for thin elastic shells

It is well known in the theory of elastic shells that a first order approximation using the shell thickness as an expansion parameter leads to the membrane theory of shells. The membrane equations have as solutions thegeneralized analytic functions. These functions have been exhaustively studied by Ilya N. Vekua [6], [7] and his students. R.P. Gilbert and J. Hile [3] introduced an extension of these systems to include elliptic systems of 2n equations in the plane and named the solutions of these systemsgeneralized hyperanalytic functions.It is shown in this paper that the next order approximation to the shell, which permits, moreover, the introduction of bending, may be described in terms of the generalized hyperanalytic functions. It is strongly suspected that the higher order approximations may also be described in terms of corresponding hypercomplex systems.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.     

On the nonlinear Riemann problems for general first elliptic systems in the plane

IntroductionManyproblemsinmechanicsandmechanicalengineeringmaybeformulatedintoboundaryvalueproblemsforfirst_orsecond_orderellipticsystems.Lotsofscholarsandtheauthorsstudiedthem ( [1～ 6] ) .InthispaperwediscussthenonlinearRiemannproblemforgeneralsystemsofthefirst_orderlinearandquasi_linearequationsintheplane.1　TheNonlinearRiemannProblemforGeneralLinearEquationLetG beamultiplyconnecteddomaininthecomplexplaneEthatisboundedbyafinitenumberofclosed ,nonintersectingC1,αcurvesΓk,k=0 ,… ,mwit…     

相空间中非保守系统的Herglotz广义变分原理及其Noether定理

与经典变分原理相比,基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述,它不仅能够描述所有采用经典变分原理能够描述的动力学过程,而且能够应用于经典变分原理不能适用的非保守或耗散系统.将Herglotz广义变分原理拓展到相空间,研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理.首先,提出相空间中Herglotz广义变分原理,给出相空间中非保守系统的变分描述,导出相应的Hamilton正则方程;其次,基于非等时变分与等时变分之间的关系,导出相空间中Hamilton-Herglotz作用量变分的两个基本公式;再次,给出Noether对称变换的定义和判据,提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理,揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系.在经典条件下,Herglotz广义变分原理退化为经典变分原理,与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理.文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性.     

Conservation laws and symmetries of systems with unilateral constraints in phase space

By introducing the generalized quasi-symmetry of the infinitestinal transformation for transformation groupG _r , this paper studies the conservation laws and symmetries of dynamical systems with unilateral constraints in phase space. Noether's theorem and Noether's inverse theorem for mechanical systems with unilateral constraints in phase space are obtained and two kinds of equivalent forms of generalized Killing equations which are used to determine the generators of the infinitestinmal group transformation are given. The project sponsored by the National Natural Science Foundation and the Doctoral Programme Foundation of Institution of Higher Education of China.     

NGPG-Stability of Linear Multistep Methods for Systems of Generalized Neutral Delay Differential Equations

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations. After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP _G -stable if and only if it is A-stable.     

一类新型正则方程对广义经典力学的推广

本文首先将一类新型正则方程推广到广义经典力学,得到广义正则方程,确定泊松括号和拉格朗日括号及它们的性质,用泊松括号表示广义正则方程,建立新型Hamilton广义变分原理、Hamilton-jacobi方程。     

EXTENSION OF THE MAC-KILLAN'S EQUATIONS TO NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this article,the Mac-Millan's equations are extended to the most general nonholonomic mechanical systems and the generalized Mac-Millan's equations for nonlinear nonholonomic systems are obtained. And then the equivalence between the generalized Mac-Millan's equations and the generalized Chaply-gin's equations is demonstrated. Finally an example is given,